SECT. IV.] EXAMINATION OF AX INTEGRAL. 429 



Now the number p being infinitely great, the second curve has 

 all its sinuosities infinitely near ; we easily see that for all points 

 which are at a finite distance from the point x, the definite 

 integral, or the whole area of the third curve, is formed of equal 

 parts alternately positive or negative, which destroy each other two 

 by two. In fact, for one of these points situated at a certain dis 

 tance from the point #, the value of /(a) varies infinitely little 



when we increase the distance by a quantity less than . The 

 same is the case with the denominator a x, which measures that 



distance. The area which corresponds to the interval is there- 



P 

 fore the same as if the quantities /(a) and a a; were not variables. 



Consequently it is nothing when a x is a finite magnitude. 

 Hence the definite integral may be taken between limits as near 

 as we please, and it gives, between those limits, the same result as 

 between infinite limits. The whole problem is reduced then to 

 taking the integral between points infinitely near, one to the left, 

 the other to the right of that where a x is nothing, that is to say 

 from OL = X co to a = x+ co, denoting by co a quantity infinitely 

 small. In this interval the function /(a) does not vary, it is 

 equal to/ (a?), and may be placed outside the symbol of integra 

 tion. Hence the value of the expression is the product of f(jc) by 



[ 



J 



a x 

 taken between the limits a x = co, and a x = co. 



Now this integral is equal to TT, as we have seen in the pre 

 ceding article ; hence the definite integral is equal to irf(x) t whence 

 we obtain the equation 



*/ \ 1 r* j s / \ ^ sin p (a. x} , . 



/&amp;lt;*) = 5z / &amp;lt;**/&amp;lt;) - &quot;irjr &amp;lt;***) 



O 



-i) ...... (B). 



J -co * &quot;&quot;CO 



417. The preceding proof supposes that notion of infinite 

 quantities which has always been admitted by geometers. It 

 would be easy to offer the same proof under another form, examin 

 ing the changes which result from the continual increase of the 



